Can high-angle annular dark field scattering be represented by a local operator?

High-angle annular dark field imaging has become an invaluable technique for recording atomic resolution STEM images. Many analyses of high-angle annular dark field images assume that the signal is the result of a local scattering operator and can be represented as a simple convolution of a probe function with a set of atomic columns. The apparent simplicity of the technique and the straightforward increase in signal with atomic number have lead to the belief that it is possible to quantify impurity concentrations at atomic column resolution. The limitations in these assumptions are examined on the basis of approximations starting from a complete theory for high-angle scattering based on multi phonon excitations. Not surprisingly, the accuracy of the local scattering operator approximation improves as the inner cut-off angle is increased.